A Transversal Property for Permutation Groups Motivated by Partial Transformations

TL;DR

This paper introduces a new property called the $(k,l)$-universal transversal property for permutation groups, classifies groups satisfying this property for $k eq 2$, and applies these results to the study of regular semigroups of partial transformations.

Contribution

It defines the $(k,l)$-universal transversal property, provides a near-complete classification for groups with this property for $k eq 2$, and explores applications to semigroup theory.

Findings

Groups with $(2,n)$-UT are exactly the primitive groups.

Groups with $(2,2)$-UT are exactly the 2-homogeneous groups.

Classification of groups satisfying $(k,l)$-UT for $k eq 2$ is nearly complete.

Abstract

In this paper we introduce the definition of $(k,l)$-universal transversal property, which is a refinement of the definition of $k$-universal transversal property, which in turn is a refinement of the classic definition of $k$-homogeneity for permutation groups. In particular, a group possesses the $(2,n)$-universal transversal property if and only if it is primitive; it possesses the $(2,2)$-universal transversal property if and only if it is $2$-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the $(k,l)$-universal transversal property, for $k\ge 3$. Then we apply this result for studying regular semigroups of partial transformations.